Ideal-Gas Approach to Hydrodynamics

Zhe-Yu Shi,Chao Gao,Hui Zhai

doi:10.1103/physrevx.11.041031

Zhe-Yu Shi, Chao Gao + Show 1 more

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https://doi.org/10.1103/physrevx.11.041031

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Abstract

Transport is one of the most important physical processes in all energy and length scales. Ideal gases and hydrodynamics are, respectively, two opposite limits of transport. Here, we present an unexpected mathematical connection between these two limits; that is, there exist situations that the solution to a class of interacting hydrodynamic equations with certain initial conditions can be exactly constructed from the dynamics of noninteracting ideal gases. We analytically provide three such examples. The first two examples focus on scale-invariant systems, which generalize fermionization to the hydrodynamics of strongly interacting systems, and determine specific initial conditions for perfect density oscillations in a harmonic trap. The third example recovers the dark soliton solution in a one-dimensional Bose condensate. The results can explain a recent puzzling experimental observation in ultracold atomic gases by the Paris group and make further predictions for future experiments. We envision that extensive examples of such an ideal-gas approach to hydrodynamics can be found by systematical numerical search, which can find broad applications in different problems in various subfields of physics.

Highlights

We envision that extensive examples of such an ideal-gas approach to hydrodynamics can be found by systematical numerical search, which can find broad applications in different problems in various subfields of physics
Studying transport of matter is an important subject in almost all subfields of physics, ranging from structure formation in astrophysics on cosmological scales [1] to collective motions of electrons in solid-state materials on microscopic scales [2], from quark-gluon plasma as the highest-temperature quantum matters created in colliders [3] to ultracold atomic gases realized at the lowest temperature in laboratories [4]
If the relaxation time τr is much shorter than the typical dynamical timescale τd, the transport is said to be in the hydrodynamic regime

Summary

INTRODUCTION

Studying transport of matter is an important subject in almost all subfields of physics, ranging from structure formation in astrophysics on cosmological scales [1] to collective motions of electrons in solid-state materials on microscopic scales [2], from quark-gluon plasma as the highest-temperature quantum matters created in colliders [3] to ultracold atomic gases realized at the lowest temperature in laboratories [4]. The main finding is that, for certain initial conditions, when the one-particle Liouville equation for ideal gases is formally recast into the form of hydrodynamic equation, the “formal pressure tensor” depends only on local density nðr; tÞ and does possess the physical meaning as the real pressure of another interacting system The second example finds out a geometric description of the specific initial conditions in two and three dimensions, under which the real space density distribution undergoes perfect breathing oscillation in a harmonic trap This provides a physical explanation for a puzzling experimental discovery in ultracold atomic gases reported by the Paris group [8]. The third example recovers the well-known dark soliton solution in one-dimensional superfluids [10,11], indicating that the connections can be found in a more broad context beyond scale-invariant systems

THE FORMALISM

EXAMPLE I

EXAMPLE II

EXAMPLE III

OUTLOOK